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Assignment (this + 2 slides) spectroscopy

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Assignment (this + 2 slides) spectroscopy
Assignment (this + 2 slides)
• Assessment of fish (cod) freshness by VIS/NIR
spectroscopy
http://unis31.unis.no/FishTime/
• MTBE Analysis by Purge and Trap GCMS
http://www.wcaslab.com/TECH/MTBE.HTM
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Goals for today …
• Spectroscopic techniques and algorithms
• Instruments and algorithms for contraband
detection
– vapor detection techniques (mostly chemistry)
– bulk detection techniques (mostly physics)
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Spectroscopies
• Signal as a function of some dispersion parameter
–
–
–
–
–
–
–
retention time (chromatographies)
drift time (ion mobility spectroscopy)
wavelength (optical spectroscopy)
frequency (NMR, NQR, ESR)
photon energy (x-ray, g-ray spectroscopies)
particle energy (photoelectron energy spectroscopy)
ion mass (mass spectroscopies)
• Always three functions, usually three modules:
– source
– dispersion element
– detector
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Principle of Conservation of Misery
• There is an inevitable tradeoff between your
ability to separate spectral components
(resolution) and your ability to detect small
quantities (sensitivity)
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Example: VIS-NIR Diffuse
Reflectance Spectrum to Measure
Fish Freshness
(probe: light in and out)
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(monochromator: specific color light out)
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What’s This GC Gizmo?
• Pipe coated (or packed with grains that are coated)
with a “sticky” liquid ...
• Inert gas (e.g., He) flows through the pipe
(“column”)
• Mixture (e.g., gasoline) squirted into “head”
• Gas (“mobile phase”) carries it over the liquid
(“stationary phase”)
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• Mixture components move at different velocities
due to different equilibria between mobile and
stationary phases
• Components emerge at column “tail”: detect with
a “universal” detector, or use as inlet to mass
spectrometer or other instrument
• MANY similar techniques: liquid
chromatography, ion mobility chromatography,
electrophoresis, and (the original) color-band
based chromatography (hence the name)
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What’s this MS Gizmo?
• Usually a separation based on mass of positive
ions; sometimes negative ions, rarely neutrals
• Usually all the ions are accelerated (and filtered)
to the same energy
• Velocity thus depends on mass: v = Sqrt(2 W/m)
• Velocity can be measured by time-of-flight, by
trajectory in a magnetic field, etc, in many
different geometries
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• Smaller lower cost alternative: quadrupole mass
spectrometers
– ions move under combined influence of DC and
oscillating (RF) electric fields; most orbits are
unbounded, but for any particular mass there is a small
region in the DC/RF amplitude plane where they are
bounded (analogous to the inverted pendulum)
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Spectroscopies:
Algorithms
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Unraveling Overlapping Spectra
• Absent separation (like GC), given the spectrum
of a mixture, how best to unravel its components
when the component spectra all overlap?
– S1 = {s11, s12, s13, ..., s1n}
1 = hexane, {1,2,3,...,n} = peak IDs
– S2 = {s21, s22, s23, ..., s2n}
2 = octane, {1,2,3,...,n} = same peak IDs
– ... etc ....
– Sm = {sm1, sm2, sm3, ..., smn}
m = Xane, {1,2,3,...,n} = same peak IDs
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• Consider the inverse
problem: given the
concentrations, it is
straightforward to
predict what the
combined spectrum
will be:
– C = {c1, c2, c3, ..., cm},
1 = hexane, 2 = octane,
..., m = Xane
– S = c1S1 + c2S2 + c3S3
+ ... + cmSm
• Or in matrix notation:
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• If we look at only as many peaks as there are
components then the matrix is square, and it is
easy: c = s-1 S
• If we have fewer peaks than components then we
are up the creek.
• If we have more peaks than components then what
to do?
• More peaks than components means we have
“extra data” that we can use to improve the
precision of our result.
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Pseudo-Inverse Method
• The trick is to multiply both sides of the equation
by sT:
– s
c
= S
(npeaks x ncomponents) (ncomponents x 1) = (npeaks x 1)
– sTs c = sTS
(ncomponents x npeaks) (npeaks x ncomponents) (ncomponents x 1)
= (ncomponents x npeaks) (npeaks x 1)
– note that sTs is square, hence it (generally) has an
inverse
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– c = (sTs)-1sTS
(ncomponents x 1) =
(ncomponents x ncomponents)-1(ncomponents x npeaks) (npeaks x 1)
– called the “pseudo-inverse method”
• Calculated component concentrations are optimal:
equivalent to least squares fitting
– i.e., algebraic least squares fit gives the same result as
matrix solution using pseudo-inverse formalism
• (Yes, of course, there are degenerate cases where
sTs doesn’t actually have an inverse, or calculating
it is unstable; then you need to use better
judgement in deciding which peaks to use!)
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Caution ...
• c = (sTs)-1sTS is the same as the optimal result you
would get if you minimized the sum of the squares
of the differences between the components of the
data set S and a “predicted” data set S = s c:
 S = Sum((sc - S)i over all npeaks spectral peaks)
dS /dcj = 0 gives ncomponents simultaneous equations
which when you solve them for {c} gives the same result
as the pseudo-inverse
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• But (to keep the notation and discussion simple)
I’ve left something out: as in our previous
discussion about how to combine multiple
measurements that have different associated
uncertainties, you need to weight each datum by a
reciprocal measure of its uncertainty, e.g., 1/si2 (in
both the least-squares and the pseudo-inverse
formulations).
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Tandem Technologies
note analogy to image processing:
not one magic bullet, but a clever
chain of simple unit operations
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Miniaturization
Ocean Optics:
optical spectrometer
optics and electronics
on a PC card; separate
light source (below),
and fiber optic (blue)
light input path
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Contraband Detection
System issues when you have to detect
something that probably isn’t there
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Pod (Probability of Detection)
FAR (False Alarm Rate)
• Illustrative problem: a town has 10 blue taxis, 90
black taxis; a man reports a hit-and-run accident
involving a blue taxi; tests show he correctly
identifies taxi color 80% of the time; what is the
probability that the taxi he saw was actually blue?
• First thought: 80%.
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• Second thought: you should ask how often he is
correct when he says he saw a blue cab. If the cab
really was blue, he reports 8 blue cabs out of 10
blue; if the cab really was black, he reports 18
blue cabs out of 80 that are actually black. So
when he reports a blue cab he is correct only
(8/(8+18)) = 31% of the time!
• (see http://www.maa.org/devlin/devlinjune.html)
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Bayes Theorem
• We start with an a priori estimate from previous
experience, etc.
Then we receive additional information from an
observation.
How do we update our estimate?
• P(blue)=0.10, P(black)=0.90 [etc., total 1., for
possibilities>2]
• P(say it is blue | if it is blue) = 0.80,
P(say it is blue | if it is black) = 0.20,
P(it is blue | if say it is blue) = ?
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Bayes Theorem
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Airport Explosives Sniffer
• P(alarm | if bomb) = 0.80 (PoD)
P(alarm | if no_bomb) = 0.01 (PFA)
P(bomb) = 0.000001
P(no_bomb) = 0.999999
• An alarm goes off; what is the probability of a real
bomb?
• P(bomb | if alarm) = P(bomb) P(alarm | if bomb)/
(P(bomb) P(alarm | if bomb) + P(no_bomb)
P(alarm | if no_bomb))
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• P(bomb | if alarm) = 0.00007994 =~= 0.00008
(false alarm rate is 99,992/100,000)
• P(bomb | if alarm) = 0.5 when P(alarm | if
no_bomb) = 0.8x10-6
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Try this one ...
• A commercial system reports NG, RDX, PETN, TNT,
Semtex, HMX.
• Terrorists use P(NG)=0.15, P(RDX)=0.10, P(PETN)=0.20,
P(TNT)=0.05, P(Semtex)=0.25, P(HMX)=0.05,
P(OTHER)=0.20.
• The instrument characteristics are P(NG_alarm | if
NG)=0.80, P(RDX_alarm | if RDX)=0.85, P(PETN_alarm
| if PETN)=0.60, P(TNT_alarm | if TNT)=0.75,
P(Semtex_alarm|if Semtex)=0.90, P(HMX_alarm|if
HMX)=0.70, P(some_alarm | if other)=0.30,
P(wrong_alarm | if any_of_the_six)=0.05, P(some_alarm |
if no_explosive)=0.01
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• One piece of luggage out of a million contains
actual explosive.
• When an alarm goes off, what is the probabability
that some explosive is actually present in the
luggage?
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